Quantum Fields and Noncommutative Spacetime

Abstract

A noncommutative structure of spacetime induced by quantum effects of gravity is proposed and the implementation of the locality principle on such a space is discussed. Heisenberg’s uncertainty relation

$$ \Delta x \geqslant \frac{\hbar } {{\Delta p}} \geqslant \frac{{\hbar c}} {E}$$

and Einstein’s formula

$$ E = mc^2 $$

imply that measurements at short scales create strong gravitational fields which eventually shield the region of interest by a horizon whose size, in the spherical symmetric case, is given by the Schwarzschild radius $R = \frac{{2Gm}} {{c^2 }}$. Hence, resolutions of distances which are smaller than the Planck length

$$ \lambda _P : = \sqrt {\frac{{G\hbar }} {{c^3 }}} = 1.6 \times 10^{ - 33} cm$$

seem to be impossible. Therefore, the association of points of some smooth manifold with the localization of events has no operative meaning.